Riemannian Geometry of $C^{1,1}$ Manifolds

Riemannian Geometry for $C^{1,1}$ manifolds contains important differences from that for $C^{2}$ manifolds. This paper develops Riemannian geometry at the $C^{1,1}$ level of regularity. It is shown that the connection is not symmetric and this leads to additional terms in curvature tensors, geodesic equations and the Bianchi identities. Failure to account for these terms leads to nonzero torsion, affecting everything from geodesics to the Einstein curvature tensor.